Def. 5. If the numerator be less than the denominator, the fraction is a 66 proper fraction:" greater, an "improper fraction."

Def. 6. A number partly integral and partly fractional, as 2 is called a "mixed number."

Def. 7. A fraction of a fraction is called a "compound fraction."

Def. 8. A "compound expression" is the name given to several quantities connected by the different signs of operations. Those quantities which compose an expression by addition or subtraction are its "terms;" those which compose an expression by multiplication are its "factors." Def. 9. A" complex fraction" is one in which numerator and denominator, one or both, are fractions, or fractional expressions.

Def. 10. A decimal fraction is one with the product of any number of tens for its denominator.

Obs. Decimal fractions are called decimals to distinguish them from other fractions called vulgar fractions, or more commonly fractions. Whenever a fraction is spoken of, a vulgar fraction is intended.

A.-Reduction of Fractions.

1. TO REDUCE A FRACTION TO ITS LOWEST TERMS.

Rule 1. Resolve, if possible, numerator and denominator into their prime factors; and divide by all the common factors.

Rule 2. Divide numerator and denominator by all the common measures, which they may have, in succession.

Rule 3. Divide numerator and denominator by their G. C. M.

Rule 4. If the numerator and denominator be exhibited in the form of the product of several factors, divide any pair of factors in numerator and denominator by any common measure. When this is done, as far as possible, multiply the remaining factors.

N.B. This last case must be particularly noticed, as containing the principle of "cancelling."

II. TO REDUCE AN IMPROPER FRACTION TO A MIXED number,

OR AN INTEGER.

Rule. Divide the numerator by the denominator; the quotient is the integer; the remainder is the numerator, and the divisor is the denominator of the fraction, which must be expressed in its lowest terms.

Or-Reduce the fraction to its lowest terms, and divide as before.

III. TO REDUCE AN INTEGER TO AN IMPROPER FRACTION WITH

A GIVEN DENOMINATOR.

Rule. Multiply the integer by the denominator; the product is the numerator of the required fraction.

EXAMPLE.

Reduce 18 to an improper fraction with denominators 3, 12.

IV. TO REDUCE FRACTIONS TO THEIR LEAST COMMON DENOMINATOR.

Rule-1. Find the L.C.M. of the denominators; this will be the L.C.D. 2. Multiply numerator and denominator of each fraction by the quotient of the L. C. D., divided by its denominator.

N.B. It is always better to let the factors of the L. C. D. appear, as then the several multipliers are readily found by casting out the factors of each denominator, and multiplying the rest.

B.-Addition of Fractions.

I. TO ADD FRACTIONS.

Rule-1. Reduce the fractions to their L. C. D.

2. Add the numerators for the numerator of the sum; which will have the L. C. D. for denominator.

3.

Reduce the result to its lowest terms.

4. If any of the fractions be mixed numbers, add the integers separately.

11. TO REDUCE A MIXED NUMBER, TO AN IMPROPER FRACTION.

Rule. Multiply the integer by the denominator, add the numerator to the product. The sum is the numerator of the fraction, which has the same denominator as before.

162336+1817

=

1824

164153

1824

C.-Subtraction of Fractions.

1. TO SUBTRACT ONE SIMPLE FRACTION FROM ANOTHER.

Rule-1. Reduce the fractions to their L. C. D.

2. Subtract the numerator of the subtrahend from that of the diminuend. 3. The difference is the numerator of remainder, and the L. C. D. the denominator.

4. Reduce the result to its lowest terms.

II. TO SUBTRACT ONE MIXED NUMBER FROM ANOTHER.

Rule-1. Reduce the fractions to their L. C. D.

2. If the numerator of the fractional subtrahend be less than that of the diminuend, subtract integer and fraction separately.

3. If the numerator of the subtrahend be greater than that of the diminuend, increase the latter by the C. D., and the integer of the subtrahend by one, and proceed as before.